Solve 0=-306x^2+528x-3216

$$0 = - 306 x ^ { 2 } + 528 x - 3216$$

Solve for x (complex solution)

$$x=\frac{44+10\sqrt{254}i}{51}\approx 0.862745098+3.124975971i$$

$$x=\frac{-10\sqrt{254}i+44}{51}\approx 0.862745098-3.124975971i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-306x^{2}+528x-3216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -306 for a, 528 for b, and -3216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-528±\sqrt{528^{2}-4\left(-306\right)\left(-3216\right)}}{2\left(-306\right)}$$

Square 528.

$$x=\frac{-528±\sqrt{278784-4\left(-306\right)\left(-3216\right)}}{2\left(-306\right)}$$

Multiply -4 times -306.

$$x=\frac{-528±\sqrt{278784+1224\left(-3216\right)}}{2\left(-306\right)}$$

Multiply 1224 times -3216.

$$x=\frac{-528±\sqrt{278784-3936384}}{2\left(-306\right)}$$

Add 278784 to -3936384.

$$x=\frac{-528±\sqrt{-3657600}}{2\left(-306\right)}$$

Take the square root of -3657600.

$$x=\frac{-528±120\sqrt{254}i}{2\left(-306\right)}$$

Multiply 2 times -306.

$$x=\frac{-528±120\sqrt{254}i}{-612}$$

Now solve the equation x=\frac{-528±120\sqrt{254}i}{-612} when ± is plus. Add -528 to 120i\sqrt{254}.

$$x=\frac{-528+120\sqrt{254}i}{-612}$$

Divide -528+120i\sqrt{254} by -612.

$$x=\frac{-10\sqrt{254}i+44}{51}$$

Now solve the equation x=\frac{-528±120\sqrt{254}i}{-612} when ± is minus. Subtract 120i\sqrt{254} from -528.

$$x=\frac{-120\sqrt{254}i-528}{-612}$$

Divide -528-120i\sqrt{254} by -612.

$$x=\frac{44+10\sqrt{254}i}{51}$$

The equation is now solved.

$$x=\frac{-10\sqrt{254}i+44}{51} x=\frac{44+10\sqrt{254}i}{51}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-306x^{2}+528x-3216=0$$

Add 3216 to both sides. Anything plus zero gives itself.

$$-306x^{2}+528x=3216$$

Divide both sides by -306.

$$\frac{-306x^{2}+528x}{-306}=\frac{3216}{-306}$$

Dividing by -306 undoes the multiplication by -306.

$$x^{2}+\frac{528}{-306}x=\frac{3216}{-306}$$

Reduce the fraction \frac{528}{-306} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{88}{51}x=\frac{3216}{-306}$$

Reduce the fraction \frac{3216}{-306} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{88}{51}x=-\frac{536}{51}$$

Divide -\frac{88}{51}, the coefficient of the x term, by 2 to get -\frac{44}{51}. Then add the square of -\frac{44}{51} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{88}{51}x+\left(-\frac{44}{51}\right)^{2}=-\frac{536}{51}+\left(-\frac{44}{51}\right)^{2}$$

Square -\frac{44}{51} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{88}{51}x+\frac{1936}{2601}=-\frac{536}{51}+\frac{1936}{2601}$$

Add -\frac{536}{51} to \frac{1936}{2601} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{88}{51}x+\frac{1936}{2601}=-\frac{25400}{2601}$$

Factor x^{2}-\frac{88}{51}x+\frac{1936}{2601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{44}{51}\right)^{2}=-\frac{25400}{2601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{44}{51}\right)^{2}}=\sqrt{-\frac{25400}{2601}}$$

Simplify.

$$x-\frac{44}{51}=\frac{10\sqrt{254}i}{51} x-\frac{44}{51}=-\frac{10\sqrt{254}i}{51}$$

Add \frac{44}{51} to both sides of the equation.

$$x=\frac{44+10\sqrt{254}i}{51} x=\frac{-10\sqrt{254}i+44}{51}$$